On the number of permutatively inequivalent basic sequences in a Banach space

摘要

Let X be a Banach space with a Schauder basis (e(n))(n is an element of N). The relation E-0 is Borel reducible to permutative equivalence between normalized block-sequences of (e(n))(n is an element of N) or X is c(o) or l(p) saturated for some 1 <= p < +infinity. If (e(n))(n is an element of N) is shrinking unconditional then either it is equivalent to the canonical basis of c(o) or l(p), 1 < p < +infinity, or the relation E-0 is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X*. If (e(n))(n is an element of N) is unconditional, then either X is isomorphic to l(2), or X contains 2(omega) subspaces or 2(omega) quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases. (C) 2006 Elsevier Inc. All rights reserved.